DoubleCategory 2009-05-16 15:57:26 2009-05-16 15:57:26 Definition internal category in $Cat$ double category \subsection{Background} Charles Ehresmann defined in 1963 a {\em double category}$\mathcal{D}$ as an internal category in the category of small categories $\bf{Cat}$. \begib{definition} A double category $mathcal{D}$ consists of: \begin{itemize} \item a set of objects, \item a set of horizontal morphisms, \item a set of vertical morphisms, and a class of squares with source and target as shown in the following diagrams: $$f: A \to B$$ A j ² A f / j ² ® B k ² C C g /D and compositions and units that satisfy axioms. \end{definition}